I had a late 80's XL600R Honda , plenty of power 100mph (on knobbys) 4valve hemi head ,primary and secondary carbs, and likely much cheaper than a 650, kick start only though. XR dirtbike I believe has the same engine also available in a 500

You could probably pick up an entire rusted out old 2wd Toyota truck for <$500 which would give you a 22R, transmission, and rear axle. Heck, you'd even have steering and brakes if you really wanted. They are incredibly easy to make so quiet all you hear are the solid cam followers clicking away.

And around here, that Toyota wouldn't even have a speck of rust on it! Nah-nah-nah-nah-naaaaaah!
(No offense intended.)

As for using an automobile, I'm thinking I would want it very small, like a Geo Metro. We want to hide the mechanical parts as much as possible. The original idea, and very popular at Burning Man for this purpose, is a golf cart.

But yeah, I'm learning about the thumpers. Most of those "Big Gun" 650s started out a bit smaller back in the 1980s, and that increases the odds of finding one cheap.

The Honda ATV is very interesting. And I could use the rear axle as an intermediate shaft to gear the thing down to Burning Man speed.

A Toyota is going to have a 3.92 first gear, 2.28 low range in the t-case if you get a 4wd, and 4.10's in the diff most likely. With small passenger car tires on the stock 15x6JJ rims I think you'd be hard pressed to go too fast.